Soit p un polynôme d'une variable complexe z. On peut trouver plusieurs inégalités reliant le module maximum de p et une combinaison de ses coefficients. Dans ce mémoire, nous étudierons principalement les preuves connues de l'inégalité de Visser. Nous montrerons aussi quelques généralisations de cette inégalité. Finalement, nous obtiendrons quelques applications de l'inégalité de Visser à l'inégalité de Chebyshev. / Let p be a polynomial in the variable z. There exist several inequalities between the coefficents of p and its maximum modulus. In this work, we shall mainly study known proofs of the Visser inquality together with some extensions. We shall finally apply the inequality of Visser to obtain extensions of the Chebyshev inequality.

We describe a fast method for the evaluation of an arbitrary high-dimensional multivariate algebraic polynomial in Chebyshev form at the nodes of an arbitrary rank-1 Chebyshev lattice. Our main focus is on conditions on rank-1 Chebyshev lattices allowing for the exact reconstruction of such polynomials from samples along such lattices and we present an algorithm for constructing suitable rank-1 Chebyshev lattices based on a component-by-component approach. Moreover, we give a method for the fast, exact and stable reconstruction.

Recent developments in the study of Majorana fermions through braid theory have shown that there exists a set of interchanges that allow for the realization of true quantum computation. Alongside these developments there have been studies of topological superconductivity which show the existence of states that exhibit non-Abelian exchange statistics. Motivated by these developments we study the differences between Abelian and non-Abelian topological phase in the vortex state through the Bogoliubov de-Gennes (BdG) formalism. Due to our interests in low-energy states we first implement computationally efficient algorithms for calculating the mean fields and computing eigenpairs in an arbitrary energy window. We have shown that these algorithms adequately reproduce results obtained from a variety of other techniques and show that these algorithms retain spatial inhomogeneity information. Our results show topological superconductivity and vortex states can coexist; providing a means to realize zero-energy bound states, the number of which corresponds to the topological phase. With the use of our methods we present results contrasting the differences between Abelian and non-Abelian topological phase. Our calculations show that an increase in Zeeman field affects numerous parameters within topological superconductors. It causes the order parameter to become more sensitive to temperature variations in addition to a reduced rate of recovery to the bulk value from a vortex core. The increased field suppresses spin-up local density of states (LDOS) in close proximity to the vortex core for low-energy states. Further, it narrows the spectral gap at the lattice centre. Both energy spectrum and LDOS calculations confirm that trivial topological phase have no zero-energy bound states, Abelian phases have an even number, while non-Abelian phases have an odd number.

New numerical methods are proposed for computing with smooth scalar and vector valued functions of two variables defined on rectangular domains. Functions are approximated to essentially machine precision by an iterative variant of Gaussian elimination that constructs near-optimal low rank approximations. Operations such as integration, differentiation, and function evaluation are particularly efficient. Explicit convergence rates are shown for the singular values of differentiable and separately analytic functions, and examples are given to demonstrate some paradoxical features of low rank approximation theory. Analogues of QR, LU, and Cholesky factorizations are introduced for matrices that are continuous in one or both directions, deriving a continuous linear algebra. New notions of triangular structures are proposed and the convergence of the infinite series associated with these factorizations is proved under certain smoothness assumptions. A robust numerical bivariate rootfinder is developed for computing the common zeros of two smooth functions via a resultant method. Using several specialized techniques the algorithm can accurately find the simple common zeros of two functions with polynomial approximants of high degree (≥ 1,000). Lastly, low rank ideas are extended to linear partial differential equations (PDEs) with variable coefficients defined on rectangles. When these ideas are used in conjunction with a new one-dimensional spectral method the resulting solver is spectrally accurate and efficient, requiring O(n²) operations for rank $1$ partial differential operators, O(n³) for rank 2, and O(n⁴) for rank &geq,3 to compute an n x n matrix of bivariate Chebyshev expansion coefficients for the PDE solution. The algorithms in this thesis are realized in a software package called Chebfun2, which is an integrated two-dimensional component of Chebfun.

A new method for transient simulation of integrated circuits has been developed and investigated. The method utilizes expansion of circuit variables into Chebyshev series. A prototype computer simulation program based on this technique has been implemented and applied in the transient simulation of several MOS circuits. The results have been compared with those generated by SPICE. The method has been also combined with the waveform relaxation technique. Several algorithms were developed using the Gauss-Seidel and Gauss-Jacobi iterative procedures. The algorithms based on the Gauss-Seidel iterative procedure were implemented in the prototype software. They offer substantial CPU time savings in comparison with SPICE without compromising the accuracy of solutions. A description of the prototype computer simulation program and a summary of the results of simulation experiments are included.

Displacement Green's function is the building block for some semi-analytical methods like Boundary Element Method (BEM), Distributed Point Source Method (DPCM), etc. In this thesis, the displacement Green`s function in anisotropic media due to a time harmonic point force is studied. Unlike the isotropic media, the Green's function in anisotropic media does not have a closed form solution. The dynamic Green's function for an anisotropic medium can be written as a summation of singular and non-singular or regular parts. The singular part, being similar to the result of static Green's function, is in the form of an integral over an oblique circular path in 3D. This integral can be evaluated either by a numerical integration technique or can be converted to a summation of algebraic terms via the calculus of residue. The other part, which is the regular part, is in the form of an integral over the surface of a unit sphere. This integral needs to be evaluated numerically and its evaluation is considerably more time consuming than the singular part. Obtaining dynamic Green's function and its spatial derivatives involves calculation of these two types of integrals. The spatial derivatives of Green's function are important in calculating quantities like stress and stain tensors. The contribution of this thesis can be divided into two parts. In the first part, different integration techniques including Gauss Quadrature, Simpson's, Chebyshev, and Lebedev integration techniques are tried out and compared for evaluation of dynamic Green’s function. In addition the solution from the residue theorem is included for the singular part. The accuracy and performance of numerical implementation is studied in detail via different numerical examples. Convergence plots are used to analyze the numerical error for both Green's function and its derivatives. The second part of contribution of this thesis relates to the mathematical derivations. As mentioned above, the regular part of dynamic Green's function, being an integral over the surface of a unit sphere, is responsible for the majority of computational time. From symmetry properties, this integration domain can be reduced to a hemisphere, but no more simplification seems to be possible for a general anisotropic medium. In this thesis, the integration domain for regular part is further reduced to a quarter of a sphere for the particular case of transversely isotropic material. This reduction proposed for the first time in this thesis nearly halves the number of integration points for the evaluation of regular part of dynamic Green's function. It significantly reduces the computational time.

Ingeniero Civil Eléctrico / Sistemas eléctricos de potencia - Administración / El cambio en la legislación referente a los requerimientos de energías renovables no convencionales (ERNC) en Chile y los precios más competitivos han incentivado fuertemente su inserción, planteando interrogantes sobre la operación técnica y económica de los sistemas eléctricos debido a la variabilidad de las ERNC y el requerimiento de redespachos intrahorarios. Ante la necesidad de mejores herramientas con resolución horaria o menor, en esta memoria de título se estudia un modelo de predespacho en base a polinomios de Chebyshev con resolución continua, permitiendo incorporar de mejor manera la variabilidad de fuentes renovables y ciertas restricciones técnicas como rampas y balances hidráulicos en comparación al método tradicional con resolución horaria. Esta formulación es fácilmente extensible a otros problemas de operación eléctrica, como microredes y coordinación hidrotérmica. En esta memoria se estudian las propiedades numéricas de los polinomios, destacando el uso de los puntos extremos de Chebyshev y el método one-side para aproximación de perfiles temporales no negativos. Posteriormente se estudia el uso de matrices operacionales de Integración y Derivación para representación de restricciones técnicas de la operación eléctrica en el continuo. Adicionalmente, se elabora un modelo uninodal y monoembalse que incorpora diversas restricciones, proponiendo una solución al tratamiento de los puntos extremos de Chebyshev en variables binarias con el fin de realizar un equivalente al modelo horario. Este modelo se compara con un predespacho horario elaborado previamente, validando así la formulación propuesta. En este trabajo, se demuestra que es posible modelar mediante polinomios de Chebyshev, obteniendo resultados de costos totales similares al modelo horario (con diferencias menores a un 0.1 %), respetando el orden de mérito, recuperando los costos marginales y valores del agua de manera equivalente. También es posible modelar mediante reducción de coeficientes (dos tercios de la totalidad de las variables de generación), pudiendo de igual forma recuperar resultados similares al despacho horario, donde fundamentalmente se recupera el mismo valor del agua, lo cual permite la extensión a modelos de coordinación hidrotérmica.

Robust algorithms for the approximation of functions are studied and developed in this thesis. Novel results and algorithms on piecewise polynomial interpolation, rational interpolation and best polynomial and rational approximations are presented. Algorithms for the extension of Chebfun, a software system for the numerical computation with functions, are described. These algorithms allow the construction and manipulation of piecewise smooth functions numerically with machine precision. Breakpoints delimiting subintervals are introduced explicitly, implicitly or automatically, the latter method combining recursive subdivision and edge detection techniques. For interpolation by rational functions with free poles, a novel method is presented. When the interpolation nodes are roots of unity or Chebyshev points the algorithm is particularly simple and relies on discrete Fourier transform matrices, which results in a fast implementation using the Fast Fourier Transform. The method is generalised for arbitrary grids, which requires the construction of polynomials orthogonal on the set of interpolation nodes. The new algorithm has connections with other methods, particularly the work of Jacobi and Kronecker, Berrut and Mittelmann, and Egecioglu and Koc. Computed rational interpolants are compared with the behaviour expected from the theory of convergence of these approximants, and the difficulties due to truncated arithmetic are explained. The appearance of common factors in the numerator and denominator due to finite precision arithmetic is characterised by the behaviour of the singular values of the linear system associated with the rational interpolation problem. Finally, new Remez algorithms for the computation of best polynomial and rational approximations are presented. These algorithms rely on interpolation, for the computation of trial functions, and on Chebfun, for the location of trial references. For polynomials, the algorithm is particularly robust and efficient, and we report experiments with degrees in the thousands. For rational functions, we clarify the numerical issues that affect its application.

This paper is a study of the relation between smoothness of the norm on a normed linear space and the property that every Chebyshev subset is convex. Every normed linear space of finite dimension, having a smooth norm, has the property that every Chebyshev subset is convex. In the second chapter two properties of the norm, uniform Gateaux differentiability and uniform Frechet differentiability where the latter implies the former, are given and are shown to be equivalent to smoothness of the norm in spaces of finite dimension. In the third chapter it is shown that every reflexive normed linear space having a uniformly Gateaux differentiable norm has the property that every weakly closed Chebyshev subset, with non-empty weak interior that is norm-wise dense in the subset, is convex.

This thesis explores how to best choose data when curve fitting using power exponential functions. The power exponential functions used are μ(b; x)=(xe1-x)b and Φ(ρ; x)=((1-x)ex)ρ . We use a number of designs such as the equidistant design, the Chebyshev design and the the D-optimal design to compare which design gives the best fit. A few examples including the logistic and the heidler function are looked at during the comparison. The measurement of the errors were made based on the sum of least squares errors in the first part and the maximum error in the second part. MATLAB was used in this comparison.